Theory of mobility edge and non-ergodic extended phase in coupled random matrices

The mobility edge, as a central concept in disordered models for localization-delocalization transitions, has rarely been discussed in the context of random matrix theory (RMT). Here we report a new class of random matrix model by direct coupling between two random matrices, showing that their overlapped spectra and un-overlapped spectra exhibit totally different scaling behaviors, which can be used to construct tunable mobility edges. This model is a direct generalization of the Rosenzweig-Porter model, which hosts ergodic, localized, and non-ergodic extended (NEE) phases. A generic theory for these phase transitions is presented, which applies equally well to dense, sparse, and even corrected random matrices in different ensembles. We show that the phase diagram is fully characterized by two scaling exponents, and they are mapped out in various conditions. Our model provides a general framework to realize the mobility edges and non-ergodic phases in a controllable way in RMT, which pave avenue for many intriguing applications both from the pure mathematics of RMT and the possible implementations of ME in many-body models, chiral symmetry breaking in QCD and the stability of the large ecosystems.